This 2-volume research monograph presents a new kind of modern differential geometry approach to such physical theories as gauge theory. Sheaf theory (geometry) and sheaf cohomology (analysis) are used to explain all the machinary of classical differential geometry without any notion of smoothness. Together they present Maxwell fields, Yang-Mills and general relativity. The text contains a treasure trove of minutely detailed and rigorous computations. This original work will appeal to physicists, mathematicians, graduate and advanced undergraduate students studying applications of differential geometry tor physical theories.